B737-800 Captain reported they began second engine start procedure without setting the brakes. The aircraft began an uncommanded forward movement before ground personnel informed the crew of the issue. The brakes were applied immediately.

Narrative

Pushback sequence began normally; but engine start sequence resulted in an aborted start. Captain directed aborted start memory items and associated QRH Checklist review inside the flight deck. At the completion of the push; it was highly likely the Pushback Driver correctly called for the parking brake to be set; but due to distractions caused by the aborted start; Captain did not accomplish this step (and could not remember acknowledging the Driver's request). Captain did ask for the Push Crew to remain with the aircraft while he and the FO troubleshot the aborted start. Seeing no obvious system malfunctions; Captain directed a second start attempt; which was successful. Captain indicated to the Tug Driver that the start was going to be successful. At starter cutout; the Tug Driver suddenly shouted; set brake; set brake; you're moving forward!" Captain applied brake pressure and abruptly stopped the aircraft's forward motion. The Tug Driver stated they were concerned we were going to make contact with them. Fortunately; a collision between aircraft and tug/towbar was avoided; and the rest of the push; start and taxi sequence was uneventful.Full credit is due to the Tug Driver for being alert to the forward movement of the aircraft. Due to the extreme darkness at push time; Captain had no sense of the aircraft moving after towbar disconnect. First Officer was not involved in this procedural error; as he was focused on monitoring engine start parameters. This incident reinforces the fact that even small distractions can disrupt the normal flow of Crew actions. Captain is entirely responsible for the failure to set and confirm the parking brake. Given the opportunity to redo the event; he would have remained focused solely on push tasks; especially setting the brake; and not allowed himself to be distracted by the looming aborted start."